NP is as easy as detecting unique solutions
Theoretical Computer Science
The Boolean hierarchy I: structural properties
SIAM Journal on Computing
The Boolean hierarchy II: applications
SIAM Journal on Computing
SIAM Journal on Computing
On unique satisfiability and the threshold behavior of randomized reductions
Journal of Computer and System Sciences
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
Tantrix TM rotation puzzles are intractable
Discrete Applied Mathematics - Fun with algorithms 2 (FUN 2001)
Complexity Theory and Cryptology
Complexity Theory and Cryptology
Tantrix: A Minute to Learn, 100 (Genetic Algorithm) Generations to Master
Genetic Programming and Evolvable Machines
IEEE Transactions on Computers
Satisfiability parsimoniously reduces to the Tantrix™ rotation puzzle problem
MCU'07 Proceedings of the 5th international conference on Machines, computations, and universality
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Holzer and Holzer [M. Holzer, W. Holzer, Tantrix(TM) rotation puzzles are intractable, Discrete Applied Mathematics 144(3) (2004) 345-358] proved that the Tantrix(TM) rotation puzzle problem with four colors is NP-complete, and they showed that the infinite variant of this problem is undecidable. In this paper, we study the three-color and two-color Tantrix(TM) rotation puzzle problems (3-TRP and 2-TRP) and their variants. Restricting the number of allowed colors to three (respectively, to two) reduces the set of available Tantrix(TM) tiles from 56 to 14 (respectively, to 8). We prove that 3-TRP and 2-TRP are NP-complete, which answers a question raised by Holzer and Holzer [M. Holzer, W. Holzer, Tantrix(TM) rotation puzzles are intractable, Discrete Applied Mathematics 144(3) (2004) 345-358] in the affirmative. Since our reductions are parsimonious, it follows that the problems Unique-3-TRP and Unique-2-TRP are DP-complete under randomized reductions. We also show that the another-solution problems associated with 4-TRP, 3-TRP, and 2-TRP are NP-complete. Finally, we prove that the infinite variants of 3-TRP and 2-TRP are undecidable.